Arithmetical ring

In algebra, a commutative ring R is said to be arithmetical (or arithmetic) if any of the following equivalent conditions hold:

1. The localization ${\displaystyle R_{\mathfrak {m}}}$ of R at ${\displaystyle {\mathfrak {m}}}$ is a uniserial ring for every maximal ideal ${\displaystyle {\mathfrak {m}}}$ of R.
2. For all ideals ${\displaystyle {\mathfrak {a}},{\mathfrak {b}}}$, and ${\displaystyle {\mathfrak {c}}}$,

   ${\displaystyle {\mathfrak {a}}\cap ({\mathfrak {b}}+{\mathfrak {c}})=({\mathfrak {a}}\cap {\mathfrak {b}})+({\mathfrak {a}}\cap {\mathfrak {c}})}$

3. For all ideals ${\displaystyle {\mathfrak {a}},{\mathfrak {b}}}$, and ${\displaystyle {\mathfrak {c}}}$,

   ${\displaystyle {\mathfrak {a}}+({\mathfrak {b}}\cap {\mathfrak {c}})=({\mathfrak {a}}+{\mathfrak {b}})\cap ({\mathfrak {a}}+{\mathfrak {c}})}$

The last two conditions both say that the lattice of all ideals of R is distributive.

An arithmetical domain is the same thing as a Prüfer domain.